To find the rate of change of price with respect to demand when the demand is 21 wigs, we need to calculate $\frac{dp}{dx}$ using the product rule. The product rule states that for functions $f(x)$ and $g(x)$, the derivative of their product is:

$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

Given: $p = (x - 2)(x^3 + 4x^2)$

Let's define:

• $f(x) = x - 2$
• $g(x) = x^3 + 4x^2$

We need to find $f'(x)$ and $g'(x)$ to use the product rule.

Step 1: Differentiate $f(x)$ and $g(x)$

1. Differentiate $f(x) = x - 2$: $f'(x) = 1$

2. Differentiate $g(x) = x^3 + 4x^2$: $g'(x) = 3x^2 + 8x$

Step 2: Apply the Product Rule

Using the product rule: $\frac{dp}{dx} = f'(x) \cdot g(x) + f(x) \cdot g'(x)$

Substitute $f(x)$, $g(x)$, $f'(x)$, and $g'(x)$: $\frac{dp}{dx} = 1 \cdot (x^3 + 4x^2) + (x - 2) \cdot (3x^2 + 8x)$

Step 3: Simplify

Expanding the expression: $\frac{dp}{dx} = x^3 + 4x^2 + (x - 2)(3x^2 + 8x)$

Now expand $(x - 2)(3x^2 + 8x)$: $(x - 2)(3x^2 + 8x) = 3x^3 + 8x^2 - 6x^2 - 16x = 3x^3 + 2x^2 - 16x$

So, $\frac{dp}{dx} = x^3 + 4x^2 + 3x^3 + 2x^2 - 16x$

Combine like terms: $\frac{dp}{dx} = 4x^3 + 6x^2 - 16x$

Step 4: Evaluate at $x = 21$

Substitute $x = 21$ into $\frac{dp}{dx}$: $\frac{dp}{dx} = 4(21)^3 + 6(21)^2 - 16(21)$

Calculate each term:

1. $4(21)^3 = 4 \cdot 9261 = 37044$
2. $6(21)^2 = 6 \cdot 441 = 2646$
3. $16(21) = 336$

So, $\frac{dp}{dx} = 37044 + 2646 - 336 = 39354$

Interpretation of the Result

When the demand is 21 wigs, the rate of change of price with respect to demand is 39354. This means that at a demand of 21 wigs, if the demand increases by 1 wig, the price per wig is expected to increase by 39354 dollars (a steep increase, suggesting strong sensitivity of price to demand).

Would you like further details on this solution?

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Here are some related questions:

1. How does the price-demand relationship affect store revenue maximization?
2. Can we analyze the price elasticity of demand based on the given equation?
3. What happens to $\frac{dp}{dx}$ as demand approaches zero?
4. How would the interpretation change if $\frac{dp}{dx}$ were negative?
5. Could we calculate the second derivative to study concavity?

Tip: To verify large calculations, break down each term carefully.